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Theorem fin23lem14 9443
Description: Lemma for fin23 9499. 𝑈 will never evolve to an empty set if it did not start with one. (Contributed by Stefan O'Rear, 1-Nov-2014.)
Hypothesis
Ref Expression
fin23lem.a 𝑈 = seq𝜔((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
Assertion
Ref Expression
fin23lem14 ((𝐴 ∈ ω ∧ ran 𝑡 ≠ ∅) → (𝑈𝐴) ≠ ∅)
Distinct variable groups:   𝑡,𝑖,𝑢   𝐴,𝑖,𝑢   𝑈,𝑖,𝑢
Allowed substitution hints:   𝐴(𝑡)   𝑈(𝑡)

Proof of Theorem fin23lem14
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6411 . . . . 5 (𝑎 = ∅ → (𝑈𝑎) = (𝑈‘∅))
21neeq1d 3030 . . . 4 (𝑎 = ∅ → ((𝑈𝑎) ≠ ∅ ↔ (𝑈‘∅) ≠ ∅))
32imbi2d 332 . . 3 (𝑎 = ∅ → (( ran 𝑡 ≠ ∅ → (𝑈𝑎) ≠ ∅) ↔ ( ran 𝑡 ≠ ∅ → (𝑈‘∅) ≠ ∅)))
4 fveq2 6411 . . . . 5 (𝑎 = 𝑏 → (𝑈𝑎) = (𝑈𝑏))
54neeq1d 3030 . . . 4 (𝑎 = 𝑏 → ((𝑈𝑎) ≠ ∅ ↔ (𝑈𝑏) ≠ ∅))
65imbi2d 332 . . 3 (𝑎 = 𝑏 → (( ran 𝑡 ≠ ∅ → (𝑈𝑎) ≠ ∅) ↔ ( ran 𝑡 ≠ ∅ → (𝑈𝑏) ≠ ∅)))
7 fveq2 6411 . . . . 5 (𝑎 = suc 𝑏 → (𝑈𝑎) = (𝑈‘suc 𝑏))
87neeq1d 3030 . . . 4 (𝑎 = suc 𝑏 → ((𝑈𝑎) ≠ ∅ ↔ (𝑈‘suc 𝑏) ≠ ∅))
98imbi2d 332 . . 3 (𝑎 = suc 𝑏 → (( ran 𝑡 ≠ ∅ → (𝑈𝑎) ≠ ∅) ↔ ( ran 𝑡 ≠ ∅ → (𝑈‘suc 𝑏) ≠ ∅)))
10 fveq2 6411 . . . . 5 (𝑎 = 𝐴 → (𝑈𝑎) = (𝑈𝐴))
1110neeq1d 3030 . . . 4 (𝑎 = 𝐴 → ((𝑈𝑎) ≠ ∅ ↔ (𝑈𝐴) ≠ ∅))
1211imbi2d 332 . . 3 (𝑎 = 𝐴 → (( ran 𝑡 ≠ ∅ → (𝑈𝑎) ≠ ∅) ↔ ( ran 𝑡 ≠ ∅ → (𝑈𝐴) ≠ ∅)))
13 vex 3388 . . . . . . 7 𝑡 ∈ V
1413rnex 7335 . . . . . 6 ran 𝑡 ∈ V
1514uniex 7187 . . . . 5 ran 𝑡 ∈ V
16 fin23lem.a . . . . . 6 𝑈 = seq𝜔((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
1716seqom0g 7790 . . . . 5 ( ran 𝑡 ∈ V → (𝑈‘∅) = ran 𝑡)
1815, 17mp1i 13 . . . 4 ( ran 𝑡 ≠ ∅ → (𝑈‘∅) = ran 𝑡)
19 id 22 . . . 4 ( ran 𝑡 ≠ ∅ → ran 𝑡 ≠ ∅)
2018, 19eqnetrd 3038 . . 3 ( ran 𝑡 ≠ ∅ → (𝑈‘∅) ≠ ∅)
2116fin23lem12 9441 . . . . . . 7 (𝑏 ∈ ω → (𝑈‘suc 𝑏) = if(((𝑡𝑏) ∩ (𝑈𝑏)) = ∅, (𝑈𝑏), ((𝑡𝑏) ∩ (𝑈𝑏))))
2221adantr 473 . . . . . 6 ((𝑏 ∈ ω ∧ (𝑈𝑏) ≠ ∅) → (𝑈‘suc 𝑏) = if(((𝑡𝑏) ∩ (𝑈𝑏)) = ∅, (𝑈𝑏), ((𝑡𝑏) ∩ (𝑈𝑏))))
23 iftrue 4283 . . . . . . . . 9 (((𝑡𝑏) ∩ (𝑈𝑏)) = ∅ → if(((𝑡𝑏) ∩ (𝑈𝑏)) = ∅, (𝑈𝑏), ((𝑡𝑏) ∩ (𝑈𝑏))) = (𝑈𝑏))
2423adantr 473 . . . . . . . 8 ((((𝑡𝑏) ∩ (𝑈𝑏)) = ∅ ∧ (𝑏 ∈ ω ∧ (𝑈𝑏) ≠ ∅)) → if(((𝑡𝑏) ∩ (𝑈𝑏)) = ∅, (𝑈𝑏), ((𝑡𝑏) ∩ (𝑈𝑏))) = (𝑈𝑏))
25 simprr 790 . . . . . . . 8 ((((𝑡𝑏) ∩ (𝑈𝑏)) = ∅ ∧ (𝑏 ∈ ω ∧ (𝑈𝑏) ≠ ∅)) → (𝑈𝑏) ≠ ∅)
2624, 25eqnetrd 3038 . . . . . . 7 ((((𝑡𝑏) ∩ (𝑈𝑏)) = ∅ ∧ (𝑏 ∈ ω ∧ (𝑈𝑏) ≠ ∅)) → if(((𝑡𝑏) ∩ (𝑈𝑏)) = ∅, (𝑈𝑏), ((𝑡𝑏) ∩ (𝑈𝑏))) ≠ ∅)
27 iffalse 4286 . . . . . . . . 9 (¬ ((𝑡𝑏) ∩ (𝑈𝑏)) = ∅ → if(((𝑡𝑏) ∩ (𝑈𝑏)) = ∅, (𝑈𝑏), ((𝑡𝑏) ∩ (𝑈𝑏))) = ((𝑡𝑏) ∩ (𝑈𝑏)))
2827adantr 473 . . . . . . . 8 ((¬ ((𝑡𝑏) ∩ (𝑈𝑏)) = ∅ ∧ (𝑏 ∈ ω ∧ (𝑈𝑏) ≠ ∅)) → if(((𝑡𝑏) ∩ (𝑈𝑏)) = ∅, (𝑈𝑏), ((𝑡𝑏) ∩ (𝑈𝑏))) = ((𝑡𝑏) ∩ (𝑈𝑏)))
29 df-ne 2972 . . . . . . . . . 10 (((𝑡𝑏) ∩ (𝑈𝑏)) ≠ ∅ ↔ ¬ ((𝑡𝑏) ∩ (𝑈𝑏)) = ∅)
3029biimpri 220 . . . . . . . . 9 (¬ ((𝑡𝑏) ∩ (𝑈𝑏)) = ∅ → ((𝑡𝑏) ∩ (𝑈𝑏)) ≠ ∅)
3130adantr 473 . . . . . . . 8 ((¬ ((𝑡𝑏) ∩ (𝑈𝑏)) = ∅ ∧ (𝑏 ∈ ω ∧ (𝑈𝑏) ≠ ∅)) → ((𝑡𝑏) ∩ (𝑈𝑏)) ≠ ∅)
3228, 31eqnetrd 3038 . . . . . . 7 ((¬ ((𝑡𝑏) ∩ (𝑈𝑏)) = ∅ ∧ (𝑏 ∈ ω ∧ (𝑈𝑏) ≠ ∅)) → if(((𝑡𝑏) ∩ (𝑈𝑏)) = ∅, (𝑈𝑏), ((𝑡𝑏) ∩ (𝑈𝑏))) ≠ ∅)
3326, 32pm2.61ian 847 . . . . . 6 ((𝑏 ∈ ω ∧ (𝑈𝑏) ≠ ∅) → if(((𝑡𝑏) ∩ (𝑈𝑏)) = ∅, (𝑈𝑏), ((𝑡𝑏) ∩ (𝑈𝑏))) ≠ ∅)
3422, 33eqnetrd 3038 . . . . 5 ((𝑏 ∈ ω ∧ (𝑈𝑏) ≠ ∅) → (𝑈‘suc 𝑏) ≠ ∅)
3534ex 402 . . . 4 (𝑏 ∈ ω → ((𝑈𝑏) ≠ ∅ → (𝑈‘suc 𝑏) ≠ ∅))
3635imim2d 57 . . 3 (𝑏 ∈ ω → (( ran 𝑡 ≠ ∅ → (𝑈𝑏) ≠ ∅) → ( ran 𝑡 ≠ ∅ → (𝑈‘suc 𝑏) ≠ ∅)))
373, 6, 9, 12, 20, 36finds 7326 . 2 (𝐴 ∈ ω → ( ran 𝑡 ≠ ∅ → (𝑈𝐴) ≠ ∅))
3837imp 396 1 ((𝐴 ∈ ω ∧ ran 𝑡 ≠ ∅) → (𝑈𝐴) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385   = wceq 1653  wcel 2157  wne 2971  Vcvv 3385  cin 3768  c0 4115  ifcif 4277   cuni 4628  ran crn 5313  suc csuc 5943  cfv 6101  cmpt2 6880  ωcom 7299  seq𝜔cseqom 7781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-seqom 7782
This theorem is referenced by:  fin23lem21  9449
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